Group #20261287 — subgroups

Subgroups — #20261287

Order 6208.

222 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15690809	C₂	1	1	yes	maximal, characteristic, complemented
2	#15690808	C₂	1	1	yes	maximal, characteristic, complemented
3	#15690800	C₂	1	2	yes	maximal, complemented
4	#15690789	C₂	1	1	yes	maximal, characteristic
5	#15690787	C₂	1	2	yes	maximal
6	Q₈×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#15690775	C₂	1	2	yes	maximal, complemented
8	#15690775	C₂	1	1	yes	maximal, characteristic, complemented
9	#15690766	C₂	1	2	yes	maximal
10	C₂×C₄×Dic₉₇	C₂	1	1	yes	maximal, characteristic
11	C₂²⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₁₉₄	C₂²	1	2	yes	—
13	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic
14	C₂²×Dic₉₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₉₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₉₇	C₂²	1	2	yes	—
17	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
18	#13095490	C₂²	1	2	yes	—
19	#13095490	C₂²	1	2	yes	—
20	#13095490	C₂²	1	2	yes	—
21	#13095490	C₂²	1	2	yes	—
22	#13095490	C₂²	1	2	yes	—
23	C₂²⋊Dic₉₇	C₂²	1	2	yes	—
24	C₄×Dic₉₇	C₂²	1	1	yes	characteristic
25	C₄×Dic₉₇	C₂²	1	1	yes	characteristic
26	C₄×Dic₉₇	C₂²	1	1	yes	characteristic
27	C₄×Dic₉₇	C₂²	1	1	yes	characteristic
28	C₄×Dic₉₇	C₂²	1	2	yes	—
29	C₄×Dic₉₇	C₂²	1	2	yes	—
30	C₂×Q₈×C₉₇	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
33	C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
34	C₂²⋊C₄×C₉₇	C₂²	1	2	yes	—
35	C₂³×C₉₇	C₂³	1	1	yes	characteristic
36	Q₈×C₉₇	—	2	2	no	—
37	C₂×C₄×C₉₇	—	2	2	no	—
38	C₂×C₄×C₉₇	C₂³	1	2	yes	—
39	C₂×C₄×C₉₇	C₂³	1	2	yes	—
40	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₉₇	—	2	2	no	—
43	C₂×Dic₉₇	Q₈	1	4	yes	complemented
44	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₉₇	C₂³	1	2	yes	—
46	C₂×Dic₉₇	C₂³	1	2	yes	—
47	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
49	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
50	Dic₁₉₄	—	2	2	no	—
51	Dic₁₉₄	—	2	4	no	—
52	C₂²×C₉₇	—	2	1	no	—
53	C₂²×C₉₇	—	2	1	no	—
54	C₂²×C₉₇	C₂×Q₈	1	2	yes	—
55	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
56	C₄×C₉₇	—	2	2	no	—
57	C₄×C₉₇	—	2	1	no	—
58	C₄×C₉₇	—	2	2	no	—
59	C₄×C₉₇	#41	1	2	yes	—
60	Dic₉₇	—	2	2	no	—
61	Dic₉₇	—	2	2	no	—
62	Dic₉₇	—	2	1	no	—
63	Dic₉₇	C₂×Q₈	1	2	yes	—
64	Dic₉₇	C₂×Q₈	1	2	yes	—
65	Dic₉₇	#41	1	2	yes	—
66	C₂×C₉₇	—	2	2	no	—
67	C₂×C₉₇	C₂²×Q₈	1	1	yes	characteristic
68	C₂×C₉₇	C₂×#41	1	1	yes	characteristic
69	C₂×C₉₇	C₂×#41	1	1	yes	characteristic
70	C₉₇	#533	1	1	yes	characteristic, complemented, Sylow
71	#533	—	97	1	no	maximal, Sylow
72	C₄⋊Q₈	—	97	1	no	—
73	C₄⋊Q₈	—	97	1	no	—
74	#125	—	97	2	no	—
75	C₂²⋊Q₈	—	97	1	no	—
76	C₂²⋊Q₈	—	97	2	no	—
77	C₂²⋊Q₈	D₉₇	1	1	yes	characteristic
78	C₄×Q₈	—	97	1	no	—
79	C₄×Q₈	—	97	1	no	—
80	C₄×Q₈	—	97	2	no	—
81	#117	—	97	2	no	—
82	C₂×C₄²	—	97	1	no	—
83	C₂×Q₈	—	97	2	no	—
84	C₂×Q₈	—	97	1	no	—
85	C₂×Q₈	C₂×D₉₇	1	1	yes	characteristic
86	C₂²×C₄	—	97	1	no	—
87	C₂²×C₄	—	97	1	no	—
88	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
89	C₄⋊C₄	—	97	2	no	—
90	C₄⋊C₄	—	97	2	no	—
91	C₄⋊C₄	—	97	2	no	—
92	C₄⋊C₄	—	97	2	no	—
93	C₄⋊C₄	—	97	2	no	—
94	C₄⋊C₄	—	97	2	no	—
95	C₄⋊C₄	—	97	1	no	—
96	C₄⋊C₄	C₂×D₉₇	1	2	yes	—
97	C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
98	C₂²⋊C₄	—	97	2	no	—
99	C₂²⋊C₄	C₂×D₉₇	1	2	yes	—
100	C₄²	—	97	1	no	—
101	C₄²	—	97	1	no	—
102	C₄²	—	97	1	no	—
103	C₄²	—	97	1	no	—
104	C₄²	—	97	2	no	—
105	C₄²	—	97	2	no	—
106	C₂³	C₂²×D₉₇	1	1	yes	characteristic
107	Q₈	—	2	2	no	—
108	Q₈	—	194	4	no	—
109	Q₈	—	194	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	97	4	no	—
112	C₂×C₄	—	97	1	no	—
113	C₂×C₄	—	97	1	no	—
114	C₂×C₄	—	97	1	no	—
115	C₂×C₄	—	97	1	no	—
116	C₂×C₄	—	97	2	no	—
117	C₂×C₄	—	97	2	no	—
118	C₂×C₄	—	194	2	no	—
119	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₉₇	1	2	yes	—
121	C₂×C₄	C₂²×D₉₇	1	2	yes	—
122	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	Q₈×D₉₇	1	2	yes	—
126	C₂²	C₂³×D₉₇	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	97	2	no	—
131	C₄	—	97	2	no	—
132	C₄	—	97	2	no	—
133	C₄	—	194	1	no	—
134	C₄	—	194	2	no	—
135	C₄	—	194	2	no	—
136	C₄	#13095510	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×Q₈×D₉₇	1	1	yes	characteristic
139	C₂	#41×D₉₇	1	1	yes	characteristic
140	C₂	C₂×#13095510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15690809

C₂

yes

maximal, characteristic, complemented

#15690808

C₂

yes

maximal, characteristic, complemented

#15690800

C₂

yes

maximal, complemented

#15690789

C₂

yes

maximal, characteristic

#15690787

C₂

yes

maximal

Q₈×Dic₉₇

C₂

yes

maximal, characteristic, complemented

#15690775

C₂

yes

maximal, complemented

#15690775

C₂

yes

maximal, characteristic, complemented

#15690766

C₂

yes

maximal

C₂×C₄×Dic₉₇

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₉₇

C₂

yes

maximal, characteristic

C₂×Dic₁₉₄

C₂²

yes

—

C₂×Dic₁₉₄

C₂²

yes

characteristic

C₂²×Dic₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

C₂²

yes

characteristic

C₄⋊Dic₉₇

C₂²

yes

—

C₄⋊Dic₉₇

C₂²

yes

characteristic

#13095490

C₂²

yes

—

#13095490

C₂²

yes

—

#13095490

C₂²

yes

—

#13095490

C₂²

yes

—

#13095490

C₂²

yes

—

C₂²⋊Dic₉₇

C₂²

yes

—

C₄×Dic₉₇

C₂²

yes

characteristic

C₄×Dic₉₇

C₂²

yes

characteristic

C₄×Dic₉₇

C₂²

yes

characteristic

C₄×Dic₉₇

C₂²

yes

characteristic

C₄×Dic₉₇

C₂²

yes

—

C₄×Dic₉₇

C₂²

yes

—

C₂×Q₈×C₉₇

C₂²

yes

characteristic

C₂²×C₄×C₉₇

C₂²

yes

characteristic

C₄⋊C₄×C₉₇

C₂²

yes

—

C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂²⋊C₄×C₉₇

C₂²

yes

—

C₂³×C₉₇

C₂³

yes

characteristic

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

Q₈

yes

complemented

C₂×Dic₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

C₂³

yes

characteristic

Dic₁₉₄

—

Dic₁₉₄

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×Q₈

yes

—

C₂²×C₉₇

C₂⁴

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#41

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

C₂×Q₈

yes

—

Dic₉₇

C₂×Q₈

yes

—

Dic₉₇

#41

yes

—

C₂×C₉₇

—

C₂×C₉₇

C₂²×Q₈

yes

characteristic

C₂×C₉₇

C₂×#41

yes

characteristic

C₂×C₉₇

C₂×#41

yes

characteristic

C₉₇

#533

yes

characteristic, complemented, Sylow

#533

—

maximal, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₉₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₉₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₉₇

yes

—

C₄⋊C₄

C₂×D₉₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₉₇

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₂³

C₂²×D₉₇

yes

characteristic

107

Q₈

—

108

Q₈

—

194

—

109

Q₈

—

194

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

194

—

119

C₂×C₄

C₂²×D₉₇

yes

characteristic

120

C₂×C₄

C₂²×D₉₇

yes

—

121

C₂×C₄

C₂²×D₉₇

yes

—

122

C₂×C₄

C₂²×D₉₇

yes

characteristic

123

C₂²

—

124

C₂²

—

125

C₂²

Q₈×D₉₇

yes

—

126

C₂²

C₂³×D₉₇

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

194

—

134

C₄

—

194

—

135

C₄

—

194

—

136

C₄

#13095510

yes

—

137

C₂

—

138

C₂

C₂×Q₈×D₉₇

yes

characteristic

139

C₂

#41×D₉₇

yes

characteristic

140

C₂

C₂×#13095510

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	C₂²⋊Q₈×C₉₇	C₂
hypercenter	C₂²⋊Q₈	D₉₇
hypocenter	C₉₇	#533
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂⁴

Fitting

C₂²⋊Q₈×C₉₇

C₂

hypercenter

C₂²⋊Q₈

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #15690809	C₂	semidirect
2. #15690808	C₂	semidirect
3. #15690800	C₂	semidirect
6. Q₈×Dic₉₇	C₂	semidirect
7. #15690775	C₂	semidirect
8. #15690775	C₂	semidirect
43. C₂×Dic₉₇	Q₈	semidirect
70. C₉₇	#533	semidirect

H ≅ G/N

type

1. #15690809

C₂

semidirect

2. #15690808

C₂

semidirect

3. #15690800

C₂

semidirect

6. Q₈×Dic₉₇

C₂

semidirect

7. #15690775

C₂

semidirect

8. #15690775

C₂

semidirect

43. C₂×Dic₉₇

Q₈

semidirect

70. C₉₇

#533

semidirect

Subgroups — #20261287

Conjugacy classes of subgroups

Complements